ieja International Electronic Journal of Algebra 1306-6048 Abdullah HARMANCI nil−INJECTIVE RINGS Wei Jun-chao Chen Jian-hua 12 01 2007 2 2 1 21 12 01 2007 Copyright © 2007, International Electronic Journal of Algebra 2007 International Electronic Journal of Algebra

A ring R is called left nil-injective if every R-homomorphism from a principal left ideal which is generated by a nilpotent element to R is a right multiplication by an element of R. In this paper, we first introduce and characterize a left nil-injective ring, which is a proper generalization of left p-injective ring. Next, various properties of left nil-injective rings are developed, many of them extend known results.

 Left minimal elements left min-abel rings strongly left min-abel rings left MC2 rings simple singular modules left nil−injective modules J.L. Chen and N.Q. Ding. On generalizations of injectivity. In International Symposium on Ring Theory (Kyongju, 1999), 85-94, Birkhauser, Boston, Boston, MA, 2001. J.L. Chen and N.Q. Ding. On regularity on rings. Algebra Colloq., 8 (2001), 267-274. [3] Jin Yong Kim. Certain rings whose simple singular modules are GP −injective. Proc. Japan Acad., 81, Ser. A (2005), 125-128. N.K. Kim, S.B. Nam and J.K. Kim. On simple singular GP −injective modules. Comm. Algebra, 27(5)(1999), 2087-2096. T.Y. Lam and Alex S. Dugas. Quasi-duo rings and stable range descent. J. Pure Appl. Algebra, 195 (2005), 243-259. S. Lang and J.L. Chen. Small injective rings. arXiv:math.RA/0505445, vl:21 May 2005. R. Yue Chi Ming. On Quasi-Frobeniusean and Artinian rings. Publications De L,institut Math´ematique, 33(47) (1983), 239-245 R.Yue Chi Ming. On p−injectivity and generalizations. Riv. Mat. Univ. Parma, (5)5 (1996), 183-188. R. Yue Chi Ming. On quasi-injectivity and Von Neumann regularity. Mountshefte fr Math., 95 (1983), 25-32. R. Yue Chi Ming. On Y J −injectivity and VNR rings. Bull. Math. Soc. Sc. Math. Roumanie Tome., 46(94)(1-2) (2003), 87-97. G.O. Michler and O.E. Villamayor. On rings whose simple modules are injec- tive. J. Algebra, 25(1973), 185-201. W.K. Nicholson and E.S. Campos. Rings with the dual of the isomorphism theorem. J. Algebra, 271 (2004), 391-406. W.K. Nicholson and J.F. Watters. Rings with projective socle. Proc. Amer. Math. Soc. 102 (1988), 443-450. W.K. Nicholson and M.F. Yousif. Minijective ring. J. Algebra, 187 (1997), 548-578. [15] W.K. Nicholson and M.F. Yousif. Weakly continuous and C2-rings. Comm. Algebra, 29(6) (2001), 2429-2466. W.K. Nicholson and M.F. Yousif. Principally injective rings. J. Algebra, 174, 77-93 (1995), 77-93. W.K. Nicholson and M.F. Yousif. On finitely embedded rings. Comm. Algebra, 28(11) (2000), 5311-5315. W.K. Nicholson and M.F. Yousif. Continuous rings and chain conditions. J. Pure Appl. Algebra, 97 (1994), 325-332. J.C. Wei. The rings characterized by minimal left ideal. Acta. Math. Sinica, 21(3) (2005), 473-482. W. Xue. A note on Y J −injectivity. Riv. Mat. Univ. Parma, (6)1 (1998), 31-37. Jun-chao Wei*and Jian-hua Chen** School of Mathematics Science, Yangzhou University, Yangzhou,225002, Jiangsu, P. R. China E-mail:*jcweiyz@yahoo.com.cn,**cjh_m@yahoo.com.cn